Chop the onion and put aside. Melt 1 tbsp. butter in a skillet and add the onion once the pan is hot. Cook on medium high until the onion begins to caramelize and turn golden brown. Remove from heat and place cooked onion in a bowl. Mix the ground chuck with the salt, pepper and cooked onion, being careful not to overmix. Shape into 4 equal patties. Set the patties aside for 15-20 minutes until grill is ready. Spray grill with grill spray and prepare a medium-hot fire in charcoal grill with lid on. Prepare the guacamole while waiting on grill to be ready. Chop the avocados into bite sized chunks. Squeeze lemon juice on top. Chop the cilantro until you have 2 teaspoons. Mix in the cilantro with a fork. Use the fork to break up the avocado until the guacamole is just a little chunky, still not completely smooth. Put in regrigerator covered until ready to assemble burgers. When grill is ready, place patties on grill and cover. Let cook on one side for 5-6 minutes. Flip them after those 5-6 minutes and top each patty with 1/4 cup of the shredded white cheddar. Put lid back on. Spread melted butter on each bun half and place on outer parts of grill and cover again. Check soon (1-2 minutes) and take rolls off when they are lightly toasted and golden brown. When the burgers have cooked for a total of 10-12 minutes, then take the patties off the grill. Cover the patties with foil. Spread the guacamole liberally on each side of the roll. Place the burger on each bottom roll, with the cheese side up. Put the guacamole slathered tops on and enjoy! Makes 4 burgers